Paper 2019/576

On Group-Characterizability of Homomorphic Secret Sharing Schemes

Reza Kaboli, Shahram Khazaei, and Maghsoud Parviz


A group-characterizable (GC) random variable is induced by a finite group, called main group, and a collection of its subgroups [Chan and Yeung 2002]. The notion extends directly to secret sharing schemes (SSS). It is known that multi-linear SSSs can be equivalently described in terms of GC ones. The proof extends to abelian SSSs, a more powerful generalization of multi-linear schemes, in a straightforward way. Both proofs are fairly easy considering the notion of dual for vector spaces and Pontryagin dual for abelian groups. However, group-characterizability of homomorphic SSSs (HSSSs), which are generalizations of abelian schemes, is non-trivial, and thus the main focus of this paper. We present a necessary and sufficient condition for a SSS to be equivalent to a GC one. Then, we use this result to show that HSSSs satisfy the sufficient condition, and consequently they are GC. Then, we strengthen this result by showing that a group-characterization can be found in which the subgroups are all normal in the main group. On the other hand, GC SSSs whose subgroups are normal in the main group can easily be shown to be homomorphic. Therefore, we essentially provide an equivalent characterization of HSSSs in terms of GC schemes. We also present two applications of our equivalent definition for HSSSs. One concerns lower bounding the information ratio of access structures for the class of HSSSs, and the other is about the coincidence between statistical, almost-perfect and perfect security notions for the same class.

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Publication info
Preprint. Minor revision.
homomorphic secret sharingaccess structureinformation theorygroup theorygroup-characterizable random variable
Contact author(s)
shahram khazaei @ gmail com
2020-04-28: last of 3 revisions
2019-05-28: received
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      author = {Reza Kaboli and Shahram Khazaei and Maghsoud Parviz},
      title = {On Group-Characterizability of Homomorphic Secret Sharing Schemes},
      howpublished = {Cryptology ePrint Archive, Paper 2019/576},
      year = {2019},
      note = {\url{}},
      url = {}
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